International Journal of Engineering and NaModel Predictive Control (MPC) techniques with several...

International Journal of Engineering and Natural Sciences (IJENS’s), Vol. 2, Num. 1

Table of Contents:

Aim and Scope

Editorial Invitation Letter

Editorial Board and Advisory Board

Analysis of the Model Predictive Current Control of the Two Level Three Phase Inverter

Gündoğan Türker Ç. ……………………………………………………...……........… p. 1 – 5

Spouted Bed Drying Characteristics of Rosehip (Rosa Canina L.)

Evin D. ……………………………………………………...………………………... p. 7 – 10

Some Curvature Properties of Generalized Complex Space Forms

Mutlu P. ……………………………..…………………………………..……….….. p. 11 – 15

Kinetic Investigation of Boronized 34CrAlNi7 Nitriding Steel

Topuz P.; Aydoğmuş T. and Aydın Ö. …………………………………………...… p. 17 – 22

Aim and Scope of IJENS’s

International Journal of Engineering and Natural Sciences (IJENS) started its publishing life

in December 2018. The aim of the journal is to serve researchers, engineers, scientists and all

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When looked at the magnitude and the impact of the scientific and technological advances

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With the goal of bringing the research conducted by Institutions and Science and Engineering

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Journal of Natural and Engineering Sciences (IJENS) three times every year. I am proud to

introduce you to the new issue of this journal on behalf of my entire team.

Prof. Dr. Feriha ERFAN KUYUMCU

Editor

PUBLISHER

Zafer Utlu, Professor


MANAGER

Gülperen Kordel


PUBLICATION COORDINATOR

Nigar Dilşat Kanat


EDITORIAL BOARD

Editor

Feriha Erfan Kuyumcu, Professor


Associate Editors

Mert Tolon, Assistant Professor


Serpil Boz, Assistant Professor


Advisory Board

Ahmet Zafer Öztürk, Professor Istanbul Gedik University

Ahmet Topuz, Professor Istanbul Arel University Arif Hepbaşlı, Professor Yaşar University Arif Karabuga, Lecturer Istanbul Gedik University

Auwal Dodo, Ph.D. Nottingham University Ayşen Demirören, Professor Istanbul Technical University

Behiye Yüksel, Associate Professor Istanbul Gedik University Bora Alboyacı, Associate Professor Kocaeli University Devrim Aydın, Assistant Professor Eastern Mediterranean University

Dilek Kurt, Associate Professor Istanbul Gedik University Fikret Tokan, Associate Professor Yıldız Technical University

Gülşen Aydın Keskin, Associate Professor Kocaeli University Güner Arkun, Professor Istanbul Gedik University

Gökhan Bulut, Associate Professor Istanbul Gedik University Hakan Yazıcı, Associate Professor Yıldız Technical University

Halil Önder, Professor Istanbul Gedik University Haslet Ekşi Koçak, Associate Professor Istanbul Gedik University

Hasila Jarimi, Ph.D. Nottingham University Hüseyin Günerhan, Associate Professor Ege University

Mehmet Ali Baykal, Professor Istanbul Gedik University

Murat Danışman, Associate Professor Istanbul Gedik University

Mustafa Koçak, Associate Professor Gedik Holding Nur Bekiroğlu, Professor Yıldız Technical University

Nuran Yörükeren, Associate Professor Kocaeli University Nurhan Türker Tokan, Associate Professor Yıldız Technical University

Nurettin Abut, Professor Kocaeli University Özden Aslan Çataltepe, Associate Professor Istanbul Gedik University

Özgen Ümit Çolak Çakır, Professor Yıldız Technical University Saffa Riffat, Professor Nottingham University

Serdar Küçük, Professor Kocaeli University Sevinç İlhan Omurca, Associate Professor Kocaeli University

Sıddık Sinan Keskin, Professor Marmara University Tarık Baykara, Professor Doğuş University

Yanan Zhang, Ph.D. Nottingham University Yate Ding, Ph.D. Nottingham University

Zafer Utlu, Professor Istanbul Gedik University Zeynep Güven Özdemir, Associate Professor Yıldız Technical University

International Journal of Engineering and Natural Sciences (IJENS), Vol. 2, Num. 1

Copyright © IJENS’s. All rights reserved. 1

Analysis of the Model Predictive Current Control

of the Two Level Three Phase Inverter

Çiğdem GÜNDOĞAN TÜRKER

Department of Mechatronic Engineering, Istanbul Gedik University

Istanbul, Turkey

[email protected]

Abstract: Model Predictive Control (MPC) Algorithms have been very popular and used widely in industrial

applications of power converters and drives. Major advantage of MPC is the flexibility to control different

variables, with constraints and additional system requirements. Also, it has been an alternative to the classical

control techniques without need of additional modulation techniques, MPC needs the proper system model in

order to calculate optimum values of the controlled variables. This paper gives an introduction about the Model

Predictive Current Algorithm. Model Predictive Current Control Algorithm is implemented for a two phase three

level drive system. After the system is modelled, the control algorithm is verified for different load condition of

an induction machine.

Key Words: Two Level Three Phase Inverter, Induction Machine, Model Predictive Control.

1. INTRODUCTION

Predictive Control techniques have been applied

in electrical machines and drive systems such as

energy, communications, medicine, mining,

transportation, etc. Most industrial applications

such as automative, space and aeronautics, railway,

ship transport, nuclear process have own particular

requirements and need electrical drives with fault-

tolerant and high reliability. With these

requirements and growing voltage levels, the

control of the multiphase converters has been

improved in last ten years [1-2].

Field-oriented control (FOC) and direct torque

control (DTC) methods are most established

methods in three-phase electrical drives control.

FOC is a modulation-based approach with a

coordinate transformation from stator fixed to a

rotor flux-oriented coordinate system. In DTC

approach, the state of the switches is selected from

a lookup table depending on the stator flux angle

and the outputs of hysteresis controllers for flux and

torque. As it is implied from the absence of a

modulator, DTC shows a faster transient response

than FOC but it has higher current, flux, and torque

ripples [4-8].

Model Predictive Control (MPC) techniques with

several advantages have been an alternative to

conventional controllers. The common property of

the Model Predictive Control Techniques is the

precalculation of the future actions of the system in

a prediction horizon time by using the system

model directly. The optimal control action is

defined according to a cost function. The system

variables are been evaluated by comparing the

reference values in a sampling time. The direct

application of the control action to the converter

without requiring a modulator is the main

advantage of MPC. Also, the cost function is an

important stage in the design of an MPC, since

required constraints and nonlinearities of the

multidimensional systems are easily implemented

and evaluated to select the optimal switching states.

However, the high switching frequency, current

ripples and computational efforts are some major

drawbacks [9-12].

This paper is organized as follows: Firstly, the

whole system which includes induction machine

driven by two level three phase inverter is described

and modelled mathematically. In section 3, Model

Predictive Control Algorithm is introduced detaily.

Finally, the simulation of the control algorithm for

the drive system is presented.

2. SYSTEM MODEL

In this study, the system is modelled for the

induction machine driven by a two level three phase

inverter. Two level three phase inverter topology

and voltage vector are shown in Figure 1. Two

semiconductor switches in each phase leg work in a

complementary manner. When the upper switch is

Gündoğan Türker Ç., (2019), Analysis of the Model Predictive Current Control of the Two Level

Three Phase Inverter

2 ISSN: 2651-5199

on with switching state ‘1’, the lower switch is off

with switching state ‘0’. There are eight possible

switching combinations for the two level three

phase inverter as the variables [ ] { } are introduced. In this way,

each phase of the two level inverter can produce

two discrete voltage levels

and

[13-14].

(a)

(b)

Figure 1. a) Topology of two level three phase

inverter, b) Voltage vector diagram.

By employing the Clarke Transformation which

the switching states are transformed from the

plane to the plane, final control set contains

only seven unique voltage vectors

[ ] .

Thus, the actual voltages applied to the windings

of the induction machine are calculated as;

(1)

The matrix K is given by;

[

√ √ ] (2)

[ ] and [ ] produces zero

voltage vectors called zero switching states,

whereas the others produce active voltage vectors

as active switching states.

Regarding the dynamics of the induction

machine, the differential equations are given in

coordinate system which is stator fixed for .

is dis

dt 1

r s- k is

kr

r 1

r- el r (3)

r r

d r

dt Lmis- k- el r r (4)

Where the coefficients are given by

and

with

,

and

.

; the fluxes, ; the currents, ; the

resistances, ; inductances, ; mutual

inductance between stator and rotor, ; the stator

voltage and ; the rotor voltage. is

the electrical angular machine speed. denotes

stator variables, denotes the rotor variables.

The stator flux can be estimated as;

(5)

The electromagnetic torque equation is given by;

(6)

The mechanical differential equation is can be

described by

(7)

3. MODEL PREDICTIVE CURRENT

CONTROL

MPC needs the proper system model in order to

calculate optimum values of the controlled

variables. The system behaviour in next sampling

interval is calculated for every switching state of

the inverter in a certain prediction horizon. MPC

determines the optimum switching states by

minimizing a cost function. A cost function is

defined according to the desired behavior of the

system including controlled variables reference

tracking by comparing the controlled variable with

its reference value. Figure 2 shows the basic control

scheme of the system [15].

Figure 2. Basic control schema for the whole

system



The predictive current controller relies on the

model of the physical drive system to predict future

stator current trajectories. The current references

and

are transformed to current references,

and

, and the controller operates in

coordinates which makes the control more

efficiently in stationary coordinates.

Conventional speed PID controller generates the

torque reference. The constant reference value of

the rotor flux magnitude is set. Based on the

reference values of the field and torque, the currents

and are produced by the equations below;

| |

(8)

| | (9)

The State-Space models of the induction machine

can be designed as;

[ is is r r ]

, u , , (10)

y is taken as the system output vector, whereas

constitutes the switching voltage vector provided

by the controller.

Based on the discrete model of system, the

current values of the controlled variables ( ) at

step k are used to predict their next values

for all N possible switching states.

In the proposed predictive algorithm, future

current is evaluated for each of the possible

seven voltage vectors which produce seven

different current predictions.

The voltage vector whose current prediction is

closest to the expected current reference

is applied to the load at the next sampling instant.

In other words, the selected vector will be the one

that minimizes the cost function.

Adding system constraints is a remarkable

feature of MPC. These constraints can be added

simply to the cost function with their specific

weighting factors. It can be implemented by an

additional term to the cost function as the distance

between the measure value of voltage at the current

state and the future state (one step time forward) as

given below;

is

-is k 1 is

-is k 1

u k 1 -u k

(11)

4. SIMULATION OF THE CONTROL

ALGORITHM

MPC algorithm for the two level inverter and

induction machine is simulated on the

Matlab/Simulink in Figure 3. The algorithm is

executed with a sampling time . The

DC link voltage is 550V. The parameters of the

induction machine is given in Table 1.

Table 1. The parameters of the induction machine.

In the simulation, the reference value of the rotor

flux magnitude is set to | | . The torque

reference is produced by the speed PI controller.

The current references are calculated as

described in the MPC algorithm.

Figure 3. The simulation blocks of MPC of the two level inverter driving the induction machine.

Gündoğan Türker Ç., (2019), Analysis of the Model Predictive Current Control of the Two Level

Three Phase Inverter

4 ISSN: 2651-5199

The stator currents at 2800rpm without load

torque are presented in Figure 4. Figure 5 shows the

stator currents when a load torque of 4 Nm is

implemented.

Figure 4. Steady state stator currents and

currents waveforms at no load and 2800rpm.

Figure 5. Steady state stator currents and

currents waveforms at 4Nm load torque and

2800rpm.

Figure 6 shows the load torque impact on the speed.

At about time 7s, 4Nm was applied to the machine

which was rotating at 2800rpm.

Figure 6. Load torque impact by changing from

0Nm to 4Nm at 2800rpm.

In Figure 7, speed reference impacts by changing

from 1500 to 2800 rpm. Figure 8 shows the current

control result by changing the speed reference.

(a)

(b)

Figure 7. a) b) stator current steps by

changing the speed from 1500 to 2800 rpm.

Figure 8. Speed reference impact, 4Nm at 2800

rpm.

5. CONCLUSION

Predictive control techniques have been a very

powerful alternative in the electric drives

applications. It is simple to apply and allows the

control of different converters without the need of

additional modulation techniques or internal

cascade control loops. The important disadvantage

of MPCs which is high calculation power is

overcome by today’s microcontrollers.

Major advantage of MPC is the flexibility to

control different variables, with constraints and

additional system requirements. This is great

potential and flexibility to improve the

performance, efficiency, and safety demanded by

the industry applications.

Model Predictive Current Control is introduced

and presented for the system consisting of a two

level inverter and an induction machine. It is



implemented in Matlab/Simulink and obtained

simulation results of the system for different load

and speed conditions. It is clearly seen that the

algorithms can track the system references without

any problems for steady state and speed steps.

REFERENCES

1. Morari, M., Lee, J.H., 1999. Model predictive

control: past,present and future,

Comp.Chem.Eng., 23, p.667-682.

2. Lee, J.H., 2011. Model Predictive Control:

Review of the three decades of development,

Int.J.Cont.Autom.Syst, 9(3), p. 415-424.

3. Linder, A., Kennel R., "Model Predictive

Control for Electric Drives", 36th Power

Electronics Specialists Conference, 2005, 1793-

1799.

4. Kazmierkowski M.P., Krishnan R., ve

Blaab erg, “Control in Power electronics,

NewYork:Academic Press, 2002.

5. Wang F., Li S., Mei X., Xie W., Rodriguez J.,

Kennel, R.M.,“Model-Based Predictive Direct

Control Strategies for Electrical Drives: An

Experimental Evaluation of PTC and PCC

Methods”, IEEE Trans. On Ind. Informations,

11,3, 2015.

6. Cortes,P.,Kazmierkowski, M.P., Kennel, R.M.,

Quevedo, D.E. ve Rodriguez, J., “Predictive

Control in Power Electronics and Drives”, IEEE

Trans. Ind. Electron., 55, 12, 4312-4324,

Dec.2008.

7. Kennel, R., Rodriguez, J., Espinoza, J.,

Trincado, M., 2010. High Performance Speed

Control Methods for Electrical Machines: An

Assessment, IEEE Int. Conf. On Industrial

Technology, p.1793-1799.

8. Buja, G.S., Kazmierkowski, M.P., Direct

Torque Control of PWM Inverter-Fed AC

motors- a Survey, IEEE Trans. on Industial

Electronics.,1793-1799.

9. Geyer, T., Papafotiou, G. And Morari, M.,

“Model Predictive Direct Torque Control-Part1:

Concept, algorithm and Analsis”, IEEE Trans.

Ind. Electron, 56, 6, 2009, 1894-1905.

10. Papafotiou, G., Kley, J., Papadopolus, K.G.,

Bohnen, P., and Morari, M., “Model Predictive

Direct Torque Control –Part II: Implementation

and E perimental Evaluation”, IEEE Trans. Ind.

Electron, 56, 6, 2009, 1906-1915

11. Geyer, T., 2014. Quevedo, D.E., "Multistep

Finite Control Set Model Predictive Control for

Power Electronics", IEE Trans. On Power

Electronics, 29, 12.

12. Scoltock, J., Geyer, T., Madawana, U., 2013. A

Comparison of Model Predictive Control

Schemes for MV Induction Motor Drives, IEEE

Trans.of Industrial Informatics, 9(2), p.909-919.

13. Rodríguez J., Pontt J., Silva C., Correa P.,

Lezana P., Cortes P., Amman U., “Predictive

Current Control of a Voltage Source Inverter

IEEE Trans. On Industrial Electronics, 54, 1,

February 2007.

14. Karamanakos, P., Stolze P., Kennel R.M.,

Manias S., Mouton H.T., “Variable Switching

Point Predictive Torque Control of Induction

Machines”, IEEE Journal of Emerging and

Selected Topics in Power Electronics, 2,2, 2014.

15. Stolze, P., Karamanakos, P., Moiton, T.,

Manias, S.N., 2013. Heuristic Variable

Switching Point Predictive Current Control for

the Three-Level Neutral Point Clamped

Inverter", SLED.



Abstract: Drying kinetics, effective moisture diffusivity and activation energy of rosehip (Rosa Canina L.) dried in a spouted bed

dryer were investigated. The effects of the spouted bed drying and the inlet air temperature in the range of 40-80°C on the

moisture ratio degradation and the drying rate of rosehip (Rosa Canina L.) were studied experimentally. Drying took place in the

falling rate period. Drying time was reduced by 83% using a temperature of 80°C instead of 40°C. The effective moisture

diffusivities of rosehip under spouted bed drying ranged from 2.5x10-10 to 2.56 x10-9 m2/s. The values of diffusivities increased

with the increase in inlet air temperature. An Arrhenius relation with an activation energy value of 51.6 kJ/mol expressed the

effect of temperature on the diffusivity.

Keywords: Spouted bed drying; rosehip; drying kinetics; effective moisture diffusivity; activation energy.

1. INTRODUCTION

In recent years, much attention has been paid to the

quality of foods during drying. Both the method of drying

and physicochemical changes that occur during drying

affects the quality of the dehydrated product [1]. Since

rosehip fruits are rich source of vitamin C and also have a

rich composition (K, P minerals and vitamin contents), they

have traditionally been used as a vitamin supplement or for

health food products in many European countries. Rosehip

extracts also possess high antioxidant capacity as well as

antimutagenic effects [2].

Spouted bed technology in solid-gas system [3] has been

proven to be an effective means of contacting for gas and

course solid particles such as Geldart type D particles [4].

Since the agitation of solids which permits the use of high

air temperature provides rapid drying without the risk of

thermal damage, drying of coarse, heat sensitive granular

materials has been the most popular application of the

spouted beds.

In present study, drying kinetics, effective moisture

diffusivity, and activation energy of rosehips dried in the

spouted bed dryer were investigated. The effects of the

spouted bed drying, the inlet air temperature and the initial

moisture content of the rosehips on the investigated

properties were discussed.

2. MATERIAL AND METHOD

2.1. Samples

Fresh rosehips (Rosa Canina L.) were harvested by hand.

They were collected in different months (September and

October) because it was also aimed to investigate the effect

of the initial moisture content of the rosehips on drying.

Rosehips approximately the same size were selected and the

average length and the diameter were measured as 2.02 and

1.125 cm, respectively. An infrared moisture analyzer

(Sartorius MA45, Germany) was used to determine the

initial moisture contents.

2.2. Drying procedure

The experimental set-up of Paraboloid Based Spouted

Bed (PBSB) dryer and the spouted bed drying mechanism

are given in Fig.1 and Fig.2, respectively. The details of the

experimental set-up and the spouted bed drying procedure

were given in a previous study of the author [5].

Figure 1. (1) Screw type compressor, (2) Air tank, (3)

Pressure gauge, (4) Air filter, (5) Pressure regulator, (6)

Rotameters, (7) Electric heater, (8) T type thermocouples,

Spouted Bed Drying Characteristics of Rosehip

(Rosa Canina L.)

Duygu Evin

Department of Mechanical Engineering, Firat University

Elazig, Turkey

[email protected]

Evin D., (2019), Spouted Bed Drying Characteristics of Rosehip (Rosa Canina L.)

8 ISSN: 2651-5199

(9) Spouted bed, (10) Data acquisition board, (11) PID

controller.

Figure 2. The spouted bed drying mechanism.

1.5 kg of rosehips were dried at 40, 70 and 80oC inlet air

temperatures with 85 m3/h air flow rate. At ten minute

intervals, rosehip samples (approximately 5 g) were

removed from the spouted bed. The moisture content of the

samples during drying was determined with an infrared

moisture analyzer (Sartorius MA45) to obtain the variation

of moisture content with drying time.

2.3. Effective moisture diffusivity

The experimental drying data for determination of

diffusivity was interpreted by using Fick’s second law.

The solution to Eq. (1) developed by Crank (1975) [6]

can be used for various regularly shaped bodies. Assuming

uniform initial moisture distribution, constant diffusion

coefficient and negligible shrinkage Eq. (2) can be

applicable for particles with cylindrical geometry.

where MR is the moisture ratio and Deff is the effective

moisture diffusivity, m2/s.

2.4. Activation energy

The factors affecting Deff are significant to clarify the

drying characteristics of a food product. Temperature is one

of the strongest factor that effects on Deff. This effect can

generally be described by an Arrhenius equation [9]:

where 0 is the Arrhenius factor s E a is the

activation energy for diffusion kJ ol is the universal

gas constant (kJ/mol.K), and T is the air temperature (K).

3. RESULTS AND DISCUSSION

3.1. Drying Kinetics

The spouted bed drying curve of rosehip in which the

moisture content decreases with the drying time is given in

Fig 3. The effect of the air temperature on drying of

rosehips can also be seen in this figure. Moisture content

decreases gradually at 40oC. On the other hand, there is a

sharp decrease in moisture content at 80oC. The drying time

required for reducing the moisture content of rosehip from

0.44 to 0.07 (g water/g dry matter) changed between 1035

and 180 min depending on the air temperature. Increasing

the air temperature in a certain air temperature range

(40-80oC in this study) accelerates the drying process.

Figure 3. The spouted bed drying curve of rosehip.

The variation of drying rate with moisture ratio which

explains the spouted bed drying behavior of rosehip is

represented by Fig. 4. The curve shows only the falling rate

period. In spouted bed drying, the whole surface of the solid

is in contact with the air, so high heat and mass transfer

coefficients cause a rapid evaporation at the surface.

Therefore, main drying takes part in the spout region.

However, moisture needs time to be transferred from the

inner part of the solid to the surface. This especially occurs

in the falling rate period at which the internal diffusion is

essential. In the annulus region of the spouted bed, the

moisture distribution of the particles are homogenized while

traveling from the top to the bottom. As can be seen from

this figure, drying rate increased with the increasing air

temperature. The highest drying rates were achieved for

spouted bed drying at 80oC inlet air temperature.

Effect of the initial moisture content on spouted bed

drying of rosehip and variation of the drying rate for 0.44

and 0.8 initial moisture contents are represented in Fig. 5

and Fig. 6, respectively. So as to investigate the effects of

initial moisture ratio on drying kinetics, a group of rosehips

were harvested in September and the other in October.


Copyright © IJENS. All rights reserved. 9

Therefore rosehips with two different initial moisture

contents (Mo) 0.44 and 0.80 (g water/ g dry matter) were

dried in the spouted bed. Increase in the initial moisture

content from 0.44 to 0.80 db increased the drying time by

55%.

Figure 4. The variation of drying rate with moisture ratio.

Figure 5. Effect of the initial moisture content on spouted

bed drying of rosehip and variation of the drying rate for

0.44 and 0.8 initial moisture contents.

Figure 6. Effect of the initial moisture content on spouted

bed drying of rosehip and variation of the drying rate for

0.44 and 0.8 initial moisture contents.

3.2. Effective moisture diffusivity and activation energy

Values of the effective moisture diffusivities of rosehip

determined by Eq.7 are given in Table 2. The effective

diffusivities of rosehip under spouted bed drying at 40-

80oC ranged from 2.5x10-10 to 2.56 x10-9 m2/s. The

values of diffusivities increased with the increase in inlet air

temperature. The determined values lie within the general

range of 10-11 to 10-9 m2/s for food materials [10].

Fig.7 shows the influence of temperature on the effective

diffusivity. The values of ln(Deff) plotted versus 1/T was

found to be essentially a straight line in the range of

temperatures indicating Arrhenius dependence.

Figure 7. The influence of temperature on the effective

diffusivity.

The activation energy of rosehip was found to be 51.6

kJ/mol. This activation energy for water diffusion in rosehip

is higher than those given in the literature for convective

drying of other foods such as; red chilli, 37.76 kJ/mol [11];

potato, 12.32-24.27 kJ/mol [12]; green bean, 35.43 kJ/mol

[13]; carrot 28.36 kJ/mol [14], and pistachio nuts, 30.79

[15], but lower than mint 82.93 kJ/mol [16] and coconut

81.11 kJ/mol [17].

4. CONCLUSION

Drying took place in the falling rate period. Increasing

the air temperature in a certain air temperature range (40-

80oC in this study) accelerated the drying process. Drying

time was reduced by 83% using a te perature of 80°C

instead of 40°C. The effective diffusivities of rosehip under

spouted bed drying at 40-80oC ranged from 2.5x10-10 to

2.56 x10-9 m2/s. The values of diffusivities increased with

the increase in inlet air temperature. The activation energy

of rosehip was found to be 51.6 kJ/mol. Increase in the

initial moisture content from 0.44 to 0.80 db increased the

drying time by 55%.

ACKNOWLEDGMENT

Financial support from the Scientific and Technological

Research Council of Turkey (TUBITAK Project number:

104M346) is gratefully acknowledged.

Evin D., (2019), Spouted Bed Drying Characteristics of Rosehip (Rosa Canina L.)

10 ISSN: 2651-5199

REFERENCES

[1] Koyuncu, T., Tosun, I. & Ustun, N. S. (2003). Drying Kinetics and

Color Retention of Dehydrated Rosehips. Drying technology, 21(7),

1369–1381.

[2] Erenturk, S., Gulaboglu, M. S. & Gultekin, S. (2005). The effects of

cutting and drying medium on the vitamin C content of rosehip

during drying. Journal of Food Engineering, 68, 513–518.

[3] Mathur, K.B. & Epstein, N. (1974). Spouted Beds. Academic Press,

New York.

[4] Geldart, D. (1986). Gas fluidization technology. Wiley, New York.

[5] Evin . Gul H. & Tanyıldızı V. 008 . Grain drying in a

paraboloid-based spouted bed with and without draft tube. Drying

Technology, 26, 1577–1583.

[6] Crank, J. The mathematics of diffusion. Oxford, UK: Oxford

University press (1975).

[7] Babalis S.J., Belessiotis V.G., Influence of the drying conditions on

the drying constants and moisture diffusivity during the thin-layer

drying of figs. Journal of Food Engineering 65 (2004) 449–458

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Abstract: The object of the present paper is to study generalized complex space forms satisfying curvature identities named

Walker type identities. Also It is proved that the difference tensor R. – .R and the Tachibana tensor Q(S, ) of any

generalized complex space form M(f1, f2) of dimensional m ≥ 4 are linearly dependent at every point of M(f1, f2). Finally

generalized complex space forms are studied under the condition R.R – Q(S,R) = L Q(g , ).

Keywords: Generalized complex space forms, Conharmonic curvature tensor, Walker type identity, Pseudosymmetric manifold,

Tachibana Tensor.

1. INTRODUCTION

In 1989, Z. Olszak has worked on the existence of a

generalized complex space form [1]. In [2 ], U.C. De and A.

Sarkar studied the nature of a generalized Sasakian space

form under some conditions regarding projective curvature

tensor. They also studied Sasakian space forms with

vanishing quasi-conformal curvature tensor and

investigated quasi-conformal flat generalized Sasakian

space forms, Ricci-symmetric and Ricci semisymmetric

generalized Sasakian space forms [3]. Venkatesha and

B.Sumangala [4], M. Atceken [5], S. Yadav and A. K.

Srivastava [6] studied on generalized space form satisfying

certain conditions on an M-projective curvature tensor,

concircular curvature and psedo projective curvature tensor

satisfying R. = 0 and many authors studied on

generalized Sasakian space form [7]. M.C. Bharathi and C.

S. Bagewadi [8] extended the study to W2 curvature,

conharmonic and concircular curvature tensors on

generalized complex space forms.

Motivated by these ideas, in the present paper, we

study generalized complex space forms satisfying curvature

identities named Walker type identities. The difference

tensor R. – .R and the Tachibana tensor Q(S, ) of any

generalized complex space form M(f1, f2) of dimensional m

≥ 4 are linearly dependent at every point of M(f1, f2).

Generalized complex space forms are studied under the

condition R. R – Q(S,R) = L Q(g, ). A Kaehler manifold is an even dimensional manifold

Mm

, where m=2n with a complex structure J and a

positive definite metric g which satisfies the following

conditions [9]

g(JX , JY) = g(X ,Y) and ,

where denotes the covariant derivative with respect to

Levi-Civita connection.

Let (M, J, g) be a Kaehler manifold with constant

holomorphic sectional curvature K(X JX) = c, then is said

to be a complex space form and it is well known that its

curvature tensor satisfies the equation

R(X,Y)Z =

{g(Y, Z)X g(X , Z)Y+ g(X,JZ)JY ( )

( ) (1)

for any vector fields X ,Y, Z on M.

An almost Hermitian manifold M is called a generalized

complex space form M(f1, f2) if its Riemannian curvature

tensor R satisfies

R(X,Y)Z = 1{g(Y, Z)X g(X , Z)Y}+f2 ( )

g(Y, JZ)JX ( ) (2)

for any vector fields X,Y,Z , where f1 and f2 are

smooth functions on M [10,11].

For a generalized complex space form M(f1, f2) we have

( ) ( ) ( ) (3)

( ) (4)

( ) , (5)

where S is the Ricci tensor, Q is the Ricci operator and r is

the scalar curvature of M(f1, f2).

Some Curvature Properties of Generalized Complex

Space Forms

Pegah Mutlu

Faculty of Engineering, Istanbul Gedik University

Kartal, 34876, Istanbul, Turkey

[email protected]

Mutlu P., (2019), Some Curvature Properties of Generalized Complex Space Forms

12 ISSN: 2651-5199

2. PRELIMINARIES

In this section, we recall some definitions and basic

formulas which will be used in the following sections.

Let (M, g) be an n-dimensional, n ≥ 3, semi-Riemannian

connected manifold of class with Levi-Civita connection

( ) being the Lie algebra of vector fields on M.

We define on M the endomorphisms X Y, (X ,Y)

and (X ,Y) of ( ) by

(X Y) Z = A(Y, Z) A(X , Z)Y, (6)

(X ,Y) Z = ( )

(X ,Y) Z = (X ,Y) Z

(8)

respectively, where A is a symmetric (0,2)-tensor on M and

X,Y,Z ( ). The Ricci tensor S, the Ricci operator Q

and the scalar curvature r of (M , g) are defined by S(X,Y)=

tr{Z ( ) , g(QX,Y) = S(X,Y) and r = tr Q, respec-

tively. [X,Y] is the Lie bracket of vector fields X and Y. In

particular we have (X Y) = X .

The Riemannian-Christoffel curvature tensor R, the

conharmonic curvature tensor and the (0,4)-tensor G of

(M, g) are defined by

R (X1, X2, X3, X4) = g ( (X1, X2) X3, X4),

(X1, X2, X3, X4) = g ( ( X1, X2) X3, X4),

(X1, X2, X3, X4) = g (( X1 X2) X3, X4),

respectively, where X1, X2, X3, X4 ( ) From (8) it follows that

( ) ( ) ,

( ) ( ) ,

( ) ( ) ,

( )+ ( ,Z,W,Y) + ( W,Y,Z) = 0.

Let (X,Y) be a skew-symmetric endomorphism of ( ) We define the (0,4)-tensor B by B(X1,X2,X3,X4) =

g( (X1,X2) X3, X4). The tensor B is said to be a generalized

curvature tensor if

B(X1, X2, X3, X4) = B(X3, X4, X1, X2),

B(X1, X2, X3, X4) + B(X2, X3, X1, X4)

+ B(X3, X1, X2, X4) = 0.

For a (0,k)-tensor field T, k ≥1, a symmetric (0,2)- tensor

field A and a generalized curvature tensor B on (M , g), we

define the (0, k+2)-tensor fields B .T and Q(A, T) by

(B .T)(X1, ..., Xk; X, Y) = T ( (X, Y)X1, X2, ..., Xk) (X1 , X2 , …, , (X,Y) Xk),

Q (A ,T) (X1, ..., Xk; X, Y) = T( ( )X1, X2, ..., Xk) (X1 , X2 , …, (X Y) Xk),

respectively, where X, Y, Z, X1, X2, ..., Xk ( )

Let (M, g) be covered by a system of charts {W; xk}. We

define by gij , Rhijk , Sij , and

= –

( +

), (9)

the local components of the metric tensor g, the

Riemannian-Christoffel curvature tensor R, the Ricci tensor

S, and the conharmonic curvature tensor , respectively.

Further, we denote by = and

.

The local components of the (0, 6)-tensor fields R.T and

Q (g ,T) on M are given by

( ) (

) ( )

( )

( )

where are the local components of the tensor T.

In this part we present some considerations leading to the

definition of Deszcz Symmetric (Pseudosymmetric in the

sense of Deszcz) and Ricci-pseudosymmetric manifolds.

A semi-Riemannian manifold (M,g) satisfying the condition

R= 0 is said to be locally symmetric. Locally symmetric

manifolds form a subclass of the class of manifolds

characterized by the condition

(12)

where is a (0,6)-tensor field with the local components

( )

(

).

Semi-Riemannian manifolds fulfilling (12) are called

semisymmetric [12]. They are not locally symmetric, in

general.

A more general class of manifolds than the class of

semisymmetric manifolds is the class of pseudosymmetric

manifolds.

A semi-Riemannian manifold (M,g) is said to be

pseudosymmetric in the sense of Deszcz [13,14] if at every

point of M the condition

( ) (13)

holds on the set = { |

( ) },

where is some function on .

A semi-Riemannian manifold (M, g) is said to be Ricci-

pseudosymmetric [15] if at every point of M the condition

R . S = Q(g, S) (14)



holds on the set = { |

}, where

is some function on Every pseudosymmetric

manifold is Ricci-pseudosymmetric. The converse

statement is not true. The class of Ricci-pseudosymmetric

manifolds is an extension of the class of Ricci-

semisymmetric (R.S= 0) manifolds as well as of the class of

pseudosymmetric manifolds. Evidently, every Ricci-

semisymmetric is Ricci-pseudosymmetric. There exist

various examples of Ricci-pseudosymmetric manifolds

which are not pseudosymmetric.

(13), (14) or other conditions of this kind are called

curvature conditions of pseudosymmetry type [16].

3. WALKER TYPE IDENTITIES ON GENERALIZED

COMPLEX SPACE FORMS

In this section, we present results on generalized complex

space forms satisfying curvature identities named Walker

type identities.

LEMMA 3.1 [17]. For a symmetric (0,2)-tensor A and a

generalized curvature tensor on a semi-Riemannian

manifold (M,g), n ≥ 3, we have

∑ ( )( )

( )( )( )

( )

It is well-known that the following identity

∑ ( )( )

( )( )( )

( )

holds on any semi-Riemannian manifold.

THEOREM 3.2. Let (M,g), n ≥ 4, be a semi-Riemannian

manifold. Then the following three equalities are

equivalent :

∑ ( )( )

( )( )( )

( )

∑ ( )( )

( )( )( )

( )

and

∑ ( – )( )

( )( )( )

(19)

on M.

Proof. In view of (10) , we have

( )

(

) ( )

( )

(

) ( )

Using (9) in (20) we obtain

( )

( )

[ ( )

+ ( ) ( )

( )] , (22)

where = .

Applying, in the same way, (9) in (21) we get

( ) ( )

( ) +

(23)

We set

[ ( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) .

Symmetrizing (22) with respect to the pairs (h,i), (j,k) and

(l,m) and applying (15) and (16) we obtain

∑ ( )( )

( )( )( )

In the same way , using (20), we have

∑ ( )( )

( )( )( )

From the last two equations we get

∑ ( )( )

( )( )( )

If , then (17) (equivalently (18), (19)) holds

on M. This completes the proof.

Mutlu P., (2019), Some Curvature Properties of Generalized Complex Space Forms

14 ISSN: 2651-5199

The equations (17) – (19) are named the Walker type

identities. We also can consider the following Walker type

identity

∑ ( )( )

( )( )( )

( )

THEOREM 3.3. Let M(f1, f2) be an m-dimensional (m ≥ 4)

generalized complex space form. Then we have

–

( ) ( )

( ) ( )

( )

( ) ( )

Proof. By using (3) the equations (22 ), (23) and (9) reduce

to

( )

( ) ( ) ( )

and

( ) ( )

respectively. Hence we have

–

( ) ( )

and so Q (g , R) = Q(g , ) . This completes the proof.

In view of the above theorem an m-dimensional (m ≥ 4)

generalized complex space form satisfying the following

conditions:

the tensors – and ( ) are linearly

dependent at every point of M(f1, f2),






dependent at every point of M(f1, f2).

COROLLARY 3.4. Let M(f1, f2), (m ≥ 4), be an m-

dimensional generalized complex space form satisfying

R. , then M(f1, f2) is semisymmetric.

THEOREM 3.4. Let M(f1, f2), be an m-dimensional (m ≥ 4)

generalized complex space form. Then the Walker type

identities (17) – (19) and (24) hold on M(f1, f2).

Proof. In view of theorem 3.3., we have

–

( ) ( )

and using (15) we get (19) (equivalently (17) and (18)).

Further, we note that

( )

This gives

(

( ) )

(

( ) )

( ) ( )

Now using (15) and (16) complete the proof.

4. GENERALIZED COMPLEX SPACE FORM

SATISFYING R. R – Q(S , R) = L Q(g , )

In this section we consider m-dimensional, (m ≥ 4),

generalized complex space forms satisfying the condition

R. R – Q(S , R) = L Q(g , ) (29)

on = ( ) , where L is some

function on .

THEOREM 4.1. Let M(f1, f2) be an m-dimensional (m ≥ 4)

generalized complex space form. If the relation (29)

fulfilled on ( ) then ( ) is pseudo-

symmetric with the function ( )

Proof. Using (3) and (28) in (29), we have

R. R – ( ) ( ) = L Q(g , )

and so

R. R ( ) ( )

This completes the proof.



ACKNOWLEDGMENT

The author would like to thank the referees for the

careful review and their valuable comments.

REFERENCES

[1] Z. Olszak, “The existance of generalized complex space form”,

Israel.J.Math., vol.65, pp.214- 218, 1989.

[2] U.C. De and A. Sarkar,“On the projective curvature tensor of

generalized Sasakian space forms”, Quaestines Mathematicae, vol.33, pp.245-252, 2010.

[3] Sarkar. A. and De. U.C, “Some curvature properties of generalized

Sasakian space forms”, Lobachevskii journal of mathematics,

Vol.33, no.1, pp.22-27, 2012.

[4] Venkatesha and B. Sumangala, “On M-Projective curvature tensor of

a generalized Sasakian space form”, Acta Math. Univ. Comenian, Vol.82(2) pp.209-217, 2013.

[5] M. Atceken , “On generalized Saskian space forms satisfying

certain conditions on the concircular curvature tensor”, Bulletin of Mathemaical analysis and applications, vol.6(1), pp.1-8, 2014.

[6] S. Yadav, D.L. Suthar and A. K. Srivastava , “Some results on M(f1,

f2, f3) 2n+1-manifolds”, Int. J. Pure Appl. Math., vol.70(3), pp.415-423, 2011.

[7] H. G. Nagaraja and Savithri Shashidhar, “On generalized Sasakian

space forms”, International Scholarly Research Network Geometry, 2012.

[8] M. C. Bharathi and C. S. Bagewadi, “On generalized complex space

forms”, IOSR Journal of Mathematics, vol.10, pp. 44-46, 2014. [9] K. Yano, Differential geometry on complex and almost complex

spaces, Pergamon Press, 1965.

[10] F. Tricerri and L. Vanhecke “Curvature tensors on almost Hermitian manifolds”, Trans. Amer. Math. Soc., vol. 267, pp.365-398, 1981.

[11] U.C. De and G.C. Ghosh “On generalized Quasi-Einstein manifolds”

Kyungpoole Math.J., vol.44, pp. 607-615, 2004. [12] Z.I. Szabó, “Structure theorems on Riemannian spaces satisfying

R(X,Y).R=0”, I. The local version, J. Differential Geom., vol.17,

pp.531-582, 1982. [13] R. Deszcz, “On pseudosymmetric spaces”, Bull. Soc. Math. Belg.

Ser., Vol. A44, pp.1-34, 1992.

[14] R. Deszcz and Ş. Yaprak, “Curvature properties of certain pseudosymmetric manifolds”,Publ. Math. Debr., vol.45, pp.334-345,

1994.

[15] R. Deszcz and M. Hotlos , “Remarks on Riemannian manifolds satisfying a certain curvature condition imposed on the Ricci tensor”,

Prace. Nauk. Pol. Szczec., vol. 11, pp. 23-34, 1989.

[16] M. Belkhelfa, R. Deszcz, M, Głogowska, M. Hotlos, D. Kowalczyk and L. Verstraelen, “ A Review on pseudosymmetry type manifolds”,

Banach Center Publ., vol.57, pp. 179-194, 2002.

[17] R. Deszcz, J. Deprez and L. Verstraelen, “Examples of pseudo-symmetric conformally flat warped products”, Chinese J. Math., vol.

17, pp.51-65, 1989.



Abstract: In this study, kinetic examinations of boronized 34CrAlNi7 Nitriding Steel samples were described. Samples were

boronized in indirect heated fluidized bed furnace consists of Ekabor 1™ boronizing agent at 1123, 1223 and 1323 K for 1, 2 and

4 hours. Morphologically and kinetic examinations of borides formed on the surface of steel samples were studied by optical

microscope, scanning electron microscope (SEM) and X-Ray diffraction (XRD). Boride layer thicknesses formed on the steel

34CrAlNi7 ranges from 46,6 ± 3,8 to 351,8 ± 15,2 µm. The hardness of the boride layer formed on the steel 34CrAlNi7 varied

between 1001 and 2896 kg/mm2. Layer growth kinetics were analyzed by measuring the extent of penetration of FeB and Fe2B

sublayers as a function of boronizing time and temperature. The kinetics of the reaction has been determined with K=Ko exp (-

Q/RT) equation. Activation energy (Q) of boronized steel 34CrAlNi7 was determined as 169 kj/mol.

Keywords: Boronizing, 34CrAlNi7, Indirect Heated Fluidized Bed Furnace, Kinetics of Boron.

1. INTRODUCTION

Boron element in the periodic table is located next to the

carbon. The boron and its compounds are in a unique

position in terms of their properties in various applications

[1]. In the periodic table, the boron, indicated by the symbol

B, is a semiconductor element with an atomic weight of

10,81 and an atomic number of 5 and it is also the first and

the lightest element of group 3A in the periodic table.

Metallic or non-metallic elements produced from boron

compounds have wide use in the industry. Under normal

conditions, boron compounds have the property of non-

metal compound, but pure boron, like carbon element, has

electrical conductivity. In addition, the crystalline boron has

similar properties to the diamond. For example, its hardness

is close to diamond [2].

Boronizing, also commonly referred to as boriding, is a

thermochemical surface hardening process applied to well

cleaned surfaces of metallic materials at high temperatures.

As a rule, Boronizing treatments are usually carried out

between 1123 and 1223 K. The boride layers formed as a

result of boronizing treatment have high hardness as well as

wear, corrosion and high heat resistance [3]. Boronizing

increases the resistance to certain acid types, partly to

hydrochloric acid. It is possible that the irregularly shaped

parts can be boronized evenly and have a positive effect on

the tool life [4]. The formation of boride layer is diffusion

controlled. As the temperature increases, the thickness of

the boride layer formed on boronized iron surfaces also

increases. The phase formed as a result of boriding of iron-

based materials, only FeB, is the permanent tension, prone

to tensile, if the phase, Fe2B, is prone to compress. Because

of this situation, the phases apply the tensile-compressive

force in the double-phase boride layers [5,6]. The hardness

depends on the type of material and FeB or Fe2B phases on

the surface. FeB phase is harder and brighter than Fe2B

phase [7]. The atoms of the boronizing compound used in

the boronizing process are settled between the atoms of the

iron-based material by diffusion. The hardness of the boride

layer changes depending on the composition of the

boronized material and the structure of the boride layer [8].

One of the methods used for the boronizing process is the

pack boronizing technique. This technique is based on the

principle of heating the material embedded in the boron

powder mixture in a heat-resistant steel pot by the furnace

[9]. There are many powder mixtures for boronizing in the

literature. But the common point of all is the formation of

boron source, activator and inert diluents. The Ekabor

boronizing agent used in this study is also a powder mixture

containing these components. It is stated that in the

literature, this boronizing agent is composed of 5% B4C +

5% KBF4 + 90% SiC [8,10].

As boronizing is widely used in different engineering

areas and industrial sectors. Some of these sectors can be

listed as follows; metallurgy and materials, mining, textile,

chemical and mechanical engineering and also agriculture,

food and porcelain industry [11].

Boriding can be carried out on different types of cast

irons and steels such as structural steels, case hardened

steels, tool steels, stainless steels, cast steels, or sintered

steels. However, due to the risk of cracking between FeB

and Fe2B phases in nitriding steels, a thick layer of boride is

not desirable [12].

Kinetic Investigation of Boronized 34CrAlNi7 Nitriding Steel

Polat TOPUZ, Tuna AYDOĞMUŞ, Özlem AYDIN

Machinery and Metal Technology Department, İstanbul Gedik University

Istanbul, Turkey

[email protected]; [email protected]; [email protected]

Topuz P. Et al., (2019), Kinetic Investigation of Boronized 34CrAlNi7 Nitriding Steel

18 ISSN: 2651-5199

In this study, the activation energy (Q) value required for

boronizing of 34CrAlNi7 nitriding steel and the growth rate

constant of boride layer were investigated. Arrhenius

equation was used to determine the relationship between

growth rate constant and activation energy [13].

2. MATERIAL AND METHOD

In this study, 34CrAlNi7 nitriding steel material was

boronized. The results of the chemical analysis performed

with optical emission spectrometry prior to the experimental

procedures are shown in Table 1. below.

Table 1. The chemical composition of 34CrAlNi7

Boronized

Material

Alloying Elements (wt.-%)

C Mn Si Cr Ni

34CrAlNi7

Nitriding

Steel

0,38 0,72 0,23 1,66 0,80

Alloying Elements (wt.-%)

Mo V W Al

0,17 0,03 0,04 0,98

The pack-boronizing method was used for the boronizing

heat treatment. In this method, commercial name is Ekabor

1™ powder mixture was used. Samples embedded in

Ekabor 1™ powder in AISI 304 stainless steel pot were

heated in fluidized bed furnace at 1123, 1223 and 1323 K as

three process temperatures and for 1h, 2h and 4h as three

different treatment times. Then the boronized samples were

cooled in air. After this processes, boronized samples were

sanded with 120 to 1000 numbered emery paper, then

polished with diamond paste. The free from scratches

samples were etched by Nital 4 (4% HNO3 + 96% ethyl

alcohol) etcher. An optical microscope and an integrated

image analyzer were used to measure the thickness of the

boron layer formed on the surface of the samples. In order

to obtain a more detailed view of the two-phase boride

layer, a SEM image of the sample was taken with the help

of the back scattered electrons. The microstructural studies

were carried out on boronized samples. Vickers hardness

tester was used for hardness measurements. The hardness

measurements were performed from surface to matrix, by 4

different points and using with 100 g. weight.

3 RESULT AND DISCUSSION

3.1. Microstructure and Hardness Analyses

It has been revealed in many studies [8, 10, 13] that the

boron layer formed on stainless steels has a columnar

morphology. On the contrary, in this study, the shape of the

boron layer formed by the saw-tooth morphology also

shown in Figures 1, 2 and 3. In addition to the binary phase

structure forming the boride layer and the matrix

microstructures are also shown in Figure 4.

1h.

2h.

3h.

Figure 1. Boride layers formed on 34CrAlNi7 at 1123 K



1h.

2h.

3h.


1h.

2h.

3h.


According to the microstructure investigations, boride

layer formed on the surface of the boronized 34CrAlNi7

nitriding steel was found to consist of FeB and Fe2B phases.

As can be seen from the SEM (BEI) image, the outermost

dark gray phase is FeB and the adjacent light gray color

phase is Fe2B.


20 ISSN: 2651-5199

Figure 4. SEM (BE) image of boronized 34CrAlNi7

In the present investigation, the boride layer thicknesses

of the boronized samples at three different temperatures and

times range from 42,8 to 367µm. Measurement results of

the layer thicknesses can be seen from Figure 3. as well as

Table 2.

.

Figure 5. Thicknesses of boride layer on 34CrAlNi7

Table 2. Boride layer thicknesses on 34CrAlNi7

The hardness values measured from the surface to the

matrix by the Vickers method at a distance of 20 µm to 220

µm and the changes in these values can be seen at Table 3.

Table 3. Microhardness measurements of the boronized

34CrAlNi7

Boronizing

Temp.

(K)

Boronizing

Time

(h.)

Microhardness Measurement

(kg/mm2)

Distances From Surface to Center

20µm 40µm 100µm 220µm

1123

4 2216 1961 426 381

2 2106 1899 389 317

1 1869 971 361 321

1223

4 2857 2446 2016 392

2 2814 2167 1896 321

1 2321 2002 1772 383

1323

4 2896 2521 2111 1997

2 2881 2186 1989 1808

1 2403 1999 1921 1001

3.2 XRD Analyses

XRD analyses were carried out on the sample that had

been boronized for 1123K, 1223K and 1323K for 1 h. For

analysis, Philips Panalytical X-Pert Pro Brand X-Ray

Diffractometer is used. Cu(Kα) having the wavelength

1.5406 Å which matches the interatomic distance of

crystalline solid materials as well as the intensity of Cu(Kα)

is higher than other which is sufficient for the diffraction of

solid material so Cu(Kα) is used for analysis. The peaks

obtained after the analysis are shown in Figure 3.

As in all steel types, boride layer formed on 34CrAlNi7

has a double-phase structure occure with FeB and Fe2B.

However, due to the alloying elements in this type steel,

small amounts of Fe3B, FeB49, Cr2B and Cr5B3 phases were

found.

(a)

(b)

Boronized

Material

Boronizing

Temperature

(K)

Boronizing

Time

(hour)

Boron

Phase

Layer

Thickness

(µm)

34CrAlNi7

1123

1 46,6±3,8

2 77,3±4,1

4 82,6±10,4

1223

1 115,2±2,1

2 138,4±3,1

4 185,5±2,1

1323

1 196,3±11,2

2 217,6±13,3

4 351,8±15,2



(c)

Figure 5. X-ray diffraction patterns of boronized

34CrAlNi7 (1h.) at different temperatures, a)1123K,

b)1223K, c)1323K

3.3 Kinetic Examinations

The equation that determines the thickness of boride

layer changes parabolically over time is given below [8,13].

(1)

According to this equation; x indicates the boride layer

thickness in cm, t indicates the boronizing time in s., and K

indicates the growth rate constant in cm2

s-1

. If the growth

kinetics of the boride layer is desired; As can be seen from

Figure 4, the square of the boride layer thickness changes

linearly over time.

Figure 6. Square of the boride layer thicknesses on

boronized steel 34CrAlNi7over treatment time.

Boron diffusion is the primary factor affecting layer

growth. The relationship between growth rate constant and

activation energy is explained by the Arrhenius equation is

given below [13,14].

(2)

According to equation; K indicates the growth rate

constant in m2

s-1

, Q indicates the activation energy in j mol-

1, K0 indicates the pre-exponential constant in m

2 s

-1 and R

indicates the gas constant in j mol-1

K-1

. Equation 3 is

natural logarithm of Equation 2.

(3)

In order to find the activation energy value, ln K - 1 / T

graph should be plot first. The slope of this graph gives the

activation energy value and this can be seen in Figure 7.

Figure 7. Growth rate constant vs. temperature of

boronized 34CrAlNi7

The activation energy and pre-exponential constant

values were obtained from the relationship of the slope of

the straight line obtained at 1 / T = 0 with the abscissa as

origin; the results are listed in Table 4.

Table 4. Activation energy, frequency factor, and diffusion

depth of boronized 34CrAlNi7

Boronized

Material

Q

(kj/mol)

K0

34CrAlNi7

169 40*10-2

x

(cm)

40*10-2

exp (20320/T) * t

3.4 Discussion

Based on these experimental results, boride layer can be

formed on the 34CrAlNi7 nitriding steel surface without

oxidation with fluidized bed furnace by pack boronizing

treatment. At the same time, it is an efficient way to obtain

high surface hardness. Increasing treatment time and

temperature, increases layer thickness.

The microstructures showed that two distinct regions

were identified on the surface of the specimens; the boride

layer formed from FeB and Fe2B phases, and matrix.

Unlike the stainless steels, the boride layer formed on

34CrAlNi7 has a saw-tooth morphology.

From the micro hardness measurements, a decrease in

hardness values from the surface to the matrix was found.

This is because, the amount of boron in the Fe2B phase is

less than in that FeB phase.


22 ISSN: 2651-5199

According to the XRD results, boride layer formed on

34CrAlNi7, has a double-phase structure occure with FeB

and Fe2B. However, due to the other alloying elements in

this type of steel, Fe3B, FeB49, Cr2B and Cr5B3 phases were

also found. The Arrhenius equation was used to calculate

the growth kinetics of the boride layer. As a result of

calculations, activation energy of boronized 34CrAlNi7

nitriding steel has been determined as 169 kJ/mol. and this

value is consistent with the other studies in the literature.

This comparison can be seen in Table 5.

Table 5. The comparison of activation energy for diffusion

of boron with respect to the different studies

Type of

steel

Range of

Temp.

(K)

Boronizing

Process

Activation

Energy

(kj/mol)

Ref.

34CrAlNi7

1123-1223 Powder

pack 270 [15]

1123-1323 Powder

pack 169

This

study

X5CrNi

18-10 1123-1323

Powder

pack 244 [16]

X5CrNi

18-10 1123-1223

Powder

pack 234 [17]

REFERENCES

[1] Narayan S. Hosmane, Boron Science: New Technologies and

Applications, 1st ed., Boca Raton, New York, USA: CRC Press,

Taylor & Francis Group, 2011, pp.1.

[2] http://www.boren.gov.tr/en/boron/boron-element, (date of

access:24.8.2018)

[3] Joseph R. Davis, Surface Hardening of Steels: Understanding the

Basics, 1st ed., Materials Park, OH, USA: ASM International, 2002,

pp.213-216.

[4] P.Topuz, E.Yılmaz and E.Gündoğdu, “Kinetic Investigations of

Boronized Cold Work Tool Steels”, Materials Testing, Vol. 56, no.

2, pp.104-110, 2014.

[5] X. Dong, J. Hu, Z. Huang, H. Wang, R. Gao, Z. Guo,“Microstructure

and Properties of Boronizing Layer of Fe-based Powder Metallurgy

Compacts Prepared by Boronizing and Sintering Simultaneously”

Science of Sintering, V.41, pp.199-207, 2009,

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[6] M.O. Domínguez, M. A. F Renteria, M. Keddam, M. E. Espinosa, O.

D. Mejia, J. I. A. Gonzalez, J. Z. Silva, S. A. M. Moreno, J. G. G.

Reyes. “Simulation of growth kinetics of Fe2B layers formed on gray

cast iron during the powder-pack boriding” MTAEC9, 48/6, 905,

2014

[7] B. M. CARUTA. Thin Films and Coatings: New Research, p.158,

Nova Publishers, 2005

[8] Sinha AK., Boriding (Boronizing) in ASM Handbook, Vol 4: Heat

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1991, pp.978-983

[9] Patric Jacquot, Advanced Techniques for Surface Engineering –

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[10] Iswadi J.,Yusof H.A.M., Rozali S. and Ogiyama H., “Effect of

powder particle sizes on the development of ultra and surface

through superplastic boronizing of duplex stainless steel” Journal of

Solid Mechanics and Materials Engineering. Vol. 1, pp.539-546,

2007.

[11] Krukovich M.G., Prusakov B.A. and Sizov I.G. The Use of Boriding

Processes in the Industrial Treatment of Details and Tools in

Plasticity of Boronized Layers. Springer Series in Materials Science,

vol 237, Springer, Cham., pp. 301-302, 2016.

[12] http://surface-heat.com/boronizing (date of access 2019)

[13] O. Özdemir, M. A.Omar, M. Usta, S. Zeytin, C.Bindal and A.H.

Ucisik, “ An investigation on boriding kinetics of AISI 316 stainless

steel”, Vacuum, Vol.83, pp.175-179, 2008.

[14] Fernandes, F. A. P., Heck, S. C., Totten, G. E. and Casteletti, L. C.,

“Wear Evaluation of PackBoronized AISI 1060 Steel”, Materials

Performance and Characterization, Vol. 2, No. 1, pp. 58–66, 2013.

[15] Efe,G.Ç.; İpek,M.; Özbek,İ.; Bindal.C: “Kinetics of Borided

31CrMoV9 and 34CrAlNi7 Steels” Materials Characterization 59,

23 – 31 ( 2008 )

[16] P. Topuz, B. Çiçek, O. Akar: “Kinetic Investigation of AISI 304

Stainless Steel Boronized in Indirect Heated Fluidized Bed Furnace”

Journal of Mining and Metallurgy, Section B: Metallurgy, Vol.52

No.(1) B, pp. 63 – 68, 2016

[17] Kayalı, Y., “Investigation of the Diffusion Kinetics of Borided

Stainless Steel”s, The Physics of Metals and Metallography, Vol.

114 (12), pp.1061–1068, 2013.

http://surface-heat.com/boronizing

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